wi4243AP/wi4244AP: Complex Analysis week 3, Monday K. P. Hart - - PowerPoint PPT Presentation

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

wi4243AP/wi4244AP: Complex Analysis

week 3, Monday

• K. P. Hart

Faculty EEMCS TU Delft

Delft, 15 September, 2014

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Outline

1

2.4: Cauchy-Riemann equations

2

2.5: Analyticity

3

2.6: Harmonic functions

4

3.1: The Exponential function

5

3.2: Trigonometric functions

6

3.3: Logarithmic functions

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Changing variables

We have x = 1

2(z + ¯

z) and y = 1

2i (z − ¯

z), so f can also be considered as a function of z and ¯ z. Apply the multi-variable chain-rule: ∂f ∂¯ z = ∂x ∂¯ z ∂f ∂x + ∂y ∂¯ z ∂f ∂y = 1 2 ∂f ∂x − 1 2i ∂f ∂y

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Changing variables

Use real and imaginary parts and the Cauchy-Riemann equations: ∂f ∂¯ z = 1 2(ux + ivx) − 1 2i (uy + ivy) = 1 2(ux − vy) − 1 2i (vx + uy) = 0 So, . . . , f is (complex) differentiable iff ∂f ∂¯ z = 0

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Analytic functions

Definition A function f is analytic at z0 if it is differentiable on some neighbourhood N(z0, ε) of z0. f is then also analytic at all points of N(z0, ε) the domain of an analytic function is open f (z) = |z|2 is differentiable at 0 but not analytic

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Orthogonal curves

Useful fact: if f = u + iv is analytic then the level curves u(x, y) = α and v(x, y) = β are always orthogonal. Use implicit differentiation dy/ dx = −ux/uy on level curves of u dy/ dx = −vx/vy on level curves of v Apply Cauchy-Riemann equations: dy dx

• u

· dy dx

• v

=

• −ux

uy

• ·
• −vx

vy

• = −ux

uy · uy ux = −1 So there.

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Example: z2

Because z2 = x2 − y2 + 2xyi we have the level curves of u = x2 − y2 (red) v = 2xy (blue) .

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Harmonic functions

Definition A function φ : R2 → R is harmonic on a domain D if it is twice differentiable and φxx(x, y) + φyy(x, y) = 0 on all of D If f = u + iv is analytic then u and v are harmonic. uxx = vyx and vxx = −uyx uyy = −vxy and vyy = uxy Now add. We will see later that all these derivatives actually exist

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Harmonic conjugate

If u and v are harmonic and such that f = u + iv is analytic then v is a harmonic conjugate of u. Equivalently: u and v satisfy ux = vy and uy = −vx — the Cauchy-Riemann equations. This is not symmetric: u is not a conjugate of v. z2 = x2 − y2 + 2xyi is analytic but iz2 = 2xy + (x2 − y2)i is not.

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Finding harmonic conjugates

If u is harmonic is there a harmonic conjugate? On simply connected domains: yes. How to find it? Force the Cauchy-Riemann equations to hold: v can be written as v(x, y) =

• −uy dx and v(x, y) =
• ux dy

Just try this!

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Example: 1

2 ln(x2 + y 2)

u(x, y) = 1

2 ln(x2 + y2) is harmonic on the right-hand half plane

(check) We have ux = x x2 + y2 and uy = y x2 + y2 Integrate v(x, y) = −

• y

x2 + y2 dx and v(x, y) =

• x

x2 + y2 dy

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Example: 1

2 ln(x2 + y 2)

We get v(x, y) =

• x

x2 + y2 dy = arctan y x + h1(x) and v(x, y) = −

• y

x2 + y2 dx = − arctan x y + h2(y) Remember: if x > 0 then arctan x + arctan 1

x = π 2 and

if x < 0 then arctan x + arctan 1

x = − π 2

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Example: 1

2 ln(x2 + y 2)

Since arctan y

x and − arctan x y differ by a constant we conclude that

v(x, y) = arctan y x + c

• n the right-hand half plane.
• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Example: 1

2 ln(x2 + y 2)

Note: 1

2 ln(x2 + y2) is harmonic on the whole complex plane,

except at (0, 0); can we define v(x, y) everywhere too? Suppose we want v(1, 1) = π

4

We must choose v(x, y) = arctan y

x on the right-hand half plane

On the upper half plane we (must) take v(x, y) = − arctan x

y + π 2

On the lower half plane we must take v(x, y) = − arctan x

y − π 2

(because v(1, −1) = − π

4 )

But now we cannot get past the negative real axis. So: simple connectivity is necessary.

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

We look at ez

Remember Definition If z = x + iy then, by definition, ez = ex(cos y + i sin y) Re ez = ex cos y and Im ez = ex sin y |ez| = ex and arg ez = y ezew = ez+w ez+2πi = ez

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

ez is an entire function

We have seen: ez is real differentiable everywhere and ux uy vx vy

• =

ex cos y −ex sin y ex sin y ex cos y

• so, by the C-R equations, it is complex differentiable everywhere.

It is an entire function (analytic on the whole complex plane). (ez)′ = ez: the matrix on the right represents multiplication by ez.

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

It is the only sensible choice

Theorem f (z) = ez is the only function that satisfies

1 it is entire 2 f ′(z) = f (z) 3 f (0) = 1

See the book for a derivation.

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Mapping behaviour

Level curves of ex cos y (red) and ex sin y (blue)

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

sin z and cos z

Definition We use the Euler formulas to define sin z = eiz − e−iz 2i and cos z = eiz + e−iz 2 Compare with hyperbolic functions: sin z = 1 i sinh iz and cos z = cosh iz

• r

sinh z = 1 i sin iz and cosh z = cos iz

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Formulas

Use addition formulas: sin(x + iy) = sin x cos iy + cos x sin iy = sin x cosh y + i cos x sinh y and cos(x + iy) = cos x cos iy − sin x sin iy = cos x cosh y − i sin x sinh y After some manipulations: | sin z| =

• sin2 x + sinh2 y and | cos z| =
• cos2 x + sinh2 y
• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Mapping behaviour of cos z

Consider w = cos z, so u = cos x cosh y and v = − sin x sinh y. If x = a is fixed then

u2 cos2 a − v2 sin2 a = 1 (hyperbola in the w-plane)

If y = b is fixed then

u2 cosh2 b + v2 sinh2 b = 1 (ellipse in the w-plane)

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Mapping behaviour of cos z

π −1 1

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

What is log z?

Definition We write w = log z to mean that ew = z. Solve this using modulus, |ew| = eu, and argument, arg ew = v. We get eu = |z|, or u = ln |z| (one value) v = arg z (many values . . . ) So, log z = ln |z| + i Arg z + 2kπi is many-valued.

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

What is Log z?

Definition The principal branch of the logarithm is Log z, given by Log z = ln |z| + i Arg z (−π < Arg z π) it is a single-valued function.

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Properties

z = eLog z is true z = Log ez is true only if −π < y π z = elog z is true, because of periodicity z = log ez does not make sense (too many right-hand sides)

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Derivative

Let’s differentiate Log z. u(x, y) = 1

2 ln(x2 + y2), so ux = x x2+y2 and uy = y x2+y2

v(x, y) = arctan y

x ( ± π), so vx = − y x2+y2 and vy = x x2+y2

The matrix 1 x2 + y2

• x

y −y x

• represents multiplication by

x−iy x2+y2 = 1 z .

So, yes: (log z)′ = 1

z .

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Heat

Consider the upper half plane and heat up the positive real axis to U Kelvin; keep the negative real axis at absolute zero. What’s the temperature distribution, T(x, y)? T(x, y) is harmonic, T(x, 0) = U (x > 0), T(x, 0) = 0 (x < 0). Take T(x, y) = U

π (π − Arg z); it’s harmonic and it does the trick.

Later: this is the only solution.

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Heat

Consider the unit disc heat up the top half of the boundary to U Kelvin; keep the other half at absolute zero. What’s the temperature distribution, T(x, y)? Map the disc onto the upper half plane by w = 1 i z − 1 z + 1 = 2y − i(x2 + y2 − 1) (x + 1)2 + y2 Check that (1, i, −1) → (0, 1, ∞).

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

Heat

Now T(x, y) = U π

• π − Arg

1 i z − 1 z + 1

• is the solution we seek.

In terms of x and y: T(x, y) = U π

• π − arctan

x2 + y2 − 1 −2y

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions

What to do?

From the book: 2.4, 2.5, 2.6, 3.1, 3.2, 3.3. Suitable exercises: 2.30 - 2.44; 3.1 –3.24 Recommended exercises: 2.30, 2.31, 2.32, 2.34, 2.35; 3.1, 3.4, 3.5, 3.6, 3.10, 3.12, 3.15 (choice), 3.17 (choice), 3.19, 3.23.

• K. P. Hart

wi4243AP/wi4244AP: Complex Analysis